Under no circumstances should you ever give a cat aspirin. It is incredibly toxic to cats. Even in small doses it can cause kidney failure, stomach ulcers, & neurological problems.
If it's not swollen, it's very unlikely it's a fracture or a break. Might be that he jumped & landed wrong & it's soft tissue damage. Watch to see if he spends a lot of time licking it; if so it's more likely there's a break in the skin (cut, bite, scratch)
I understand sometimes we get into circumstances where a visit to the vet is nigh on impossible, I've been there. Do watch to make sure there is no swelling, that he eats, drinks & relieves himself normally. I know you hate to see your little guy in pain, but you should never, ever medicate a cat without veterinary supervision. It's just to likely you'd poison him.